The fourier-transform tag has no usage guidance.

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### Sobolev norm of a composition with a singular homeo

Let $H_p^t(\mathbb{R})$ be a fractional Sobolev space with the standard norm. The with $p>1$, $0<t<1$. Take some smooth $\phi$ from this space. Let $T$ be an ivertible homeomorphism of ...

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70 views

### Green's Function for a Kernel with Symmetric Fourier Transform $\nabla^2-x^2$

I am trying to find the inverse of the following kernel in 3 dimensions
$$
\nabla^2-x^2,
$$
where,
$$
x^2=\vec{x}.\vec{x}
$$
It seems quit simple and one would think there should already be solutions ...

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15 views

### amplitude at exact frequency in wide band signal [closed]

Could anyone suggest the most computationaly efficient method for finding amplitude of exact frequency having a noisy wide band signal.
To be more specific about a task. I have some physical ...

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62 views

### Solution to inhomogenous PDE

Given the equation $(1-\Delta)u=f$ for $f \in S(\mathbb{R}^n)$ (rapidly decreasing functions) we get by taking the Fourier transform that
$u = ...

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**1**answer

180 views

### Fourier transform surjective on $L^p(\mathbb{R}^n)$ for $p \in (1,2)$?

I know that $F_2:L^2 \rightarrow L^2$ is of course unitary, whereas $F_1:L^1 \rightarrow C_0$ is injective but not surjective. This can be seen by looking at the dual map.
Riesz-Thorin gives us that ...

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1k views

### Eigenvectors of the Fourier transformation

The Fourier transform $\hat u$ is defined on the Schwartz space $\mathscr S(\mathbb R^n)$
by
$
\hat u(\xi)=\int e^{-2iπ x\cdot \xi} u(x) dx.
$
It is an isomorphism of $\mathscr S(\mathbb R^n)$ and the ...

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28 views

### Fourier tranform of the Euclidean norm [migrated]

where can I find the Fourier transform of the power of the Euclidean norm?, that is:
$$\mathcal{F}[\|x\|^{p}](\omega) = \int_{\mathbb{R}^{d}}\exp(-2\pi i \langle\omega, x\rangle) \|x\|^{p} dx$$
...

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482 views

### Is there a $C_c^{\infty}( \mathbb{R}^d)$ function whose Fourier transform we can explicitly write down?

I noticed that although $C_c^{\infty}$-functions are dense in some quite large spaces and well understood (especially their Fourier transform) I have never encountered an explicit example of a ...

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**1**answer

400 views

### What is this Lie algebra?

Consider two matrices $A,B \in \mathfrak{su}(N)$ which are both diagonal in the standard basis and non-zero.
If we consider the new matrix $\tilde{B} := FBF^{\dagger}$ where $F$ is the `quantum' ...

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**1**answer

54 views

### Restricted Isometry Property for Discrete Fourier Transform Matrix

I was wondering if the Restricted Isometry Property holds for Discrete Fourier Transform. In particular, I am interested in whether a subsampled DFT matrix has such property. Let$W \in ...

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**1**answer

87 views

### Estimate a Fourier Transform [closed]

I'm reading an article which claims the following result (p.9): if $f : \mathbb{R}^{2} \to \mathbb{R}$ is of the form $f(x_1,x_2) = \sin (N x_{1}) h (g^{-1}(x))$, where $g$ is a diffeomorphism and $h$ ...

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votes

**1**answer

49 views

### How to relate this summation to standard discrete cosine transformation?

The standard type III discrete cosine transformation (DCT) is defined as follows:
$${X_k} = \frac{1}{2}{x_0} + \sum\limits_{n = 1}^{N - 1} {{x_n}} \cos \left[ {\frac{\pi }{N}n\left( {k + \frac{1}{2}} ...

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150 views

### Determining convexity of a polygon from its Fourier coefficients

Consider an $n$-sided polygonal curve in the plane, represented by an ordered set of points $(x_0, x_1, \ldots, x_{n-1})$; line segments connect consecutive points and also $x_{n-1}$ to $x_0$. It is ...

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**1**answer

41 views

### Subquadratic multiplication of probability mass functions (with log-convolution?)

We are currently looking for a fast, i.e. subquadratic, algorithm for the following equation:
$z_m = \sum_{i,j :\, (i \cdot j) = m} x_i \cdot y_j$.
That is, we are given two finite input vectors $x$ ...

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**1**answer

108 views

### What are the spaces for which the Fourier transform is an automorphism? [closed]

this is well-known that the Fourier transform is an automorphism of $L^2(\mathbb R)$ and also of $\mathcal S(\mathbb R)$ (Schwartz space). Is there any other spaces of functions of one real variable ...

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288 views

### Why decompose a function with eigenvectors of Laplace operator? [closed]

On periodic domain, people always use Fourier basis, which eigenvectors of Laplace operator. On sphere, people use spherical harmonics, which also are eigenvectors of Laplace operator. In applied ...

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65 views

### Fourier transform in Lorentz spaces

Denote by $E$ the class of nonnegative, even functions on $\mathbb{R}$, monotone
decreasing to 0 on $\mathbb{R}_+$.
In here, Y. Sagher stated the following proposition about Fourier transform of ...

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103 views

### If $\mathcal{F}$ is the Fourier transform, what can be said about $\mathcal{F}(L^1(\mathbb{R})) \cap L^1(\mathbb{R})$?

The Fourier transform gives a map of the Schwartz space to itself which turns out to be a linear homeomorphism of period 4.
However, when the domain is extended to $L^1(\mathbb{R})$, the situation is ...

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173 views

### Can we do better than zero padding of FFT?

My background is in signal processing, and never took any course related to functional analysis or even advanced algebra. But I have a strong conviction (may be wrong) that we may be do better then ...

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82 views

### Resolvent of the operator

Consider the Laplace operator defined on the biggest possible subset of$L^{2}(R^{2})$:
$T= - \partial^{2}_{x} -\partial^{2}_{y}+x^{2}+y^{2}+ 2.i(x \frac{\partial}{\partial ...

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61 views

### Conceptual explanation for Poisson summation formula

The Poisson summation formula says that for a Schwartz function $f : \mathbf R^d \to \mathbf R$ and its Fourier transform $\widehat f$, we have
$$\sum_{n \in \mathbf Z^d} f(x) = \sum_{n \in \mathbf ...

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**1**answer

210 views

### Solving a simple Schrödinger equation with Fast Fourier Transforms

While trying to solve a stochastic Gross-Pitaevskii equation I have found a problem that can be tracked down to something buggy occurring in the simplest Schrodinger equation possible:
$$\partial_t ...

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321 views

### Are the zeroes of the Fourier Transform of compactly supported functions isolated?

I have a continuous function $f$ on a locally compact Abelian group $G$ with compact support, and I would like to say that the zeroes of $f$ are sparse in some sense (isolated would be good, uniformly ...

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162 views

### Is this function Schwartz?

I already asked this question here on MSE, didn't get an answer, and I'm still stuck with it.
Suppose I have a smooth function $\psi$ from $\mathbb{R}^n$ to $\mathbb{C}$, for which I know that
$$
...

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267 views

### G-Correlation of Vectors

Let $\vec{a},\vec{b} \in \mathbb{R}^{n}$. Consider the function $f: S_n \to \mathbb{R}$ given by $f(\sigma):= \sum_{i=1}^{n} a_i b_{\sigma(i)}$. Let $G$ be a subgroup of $S_n$, given by $O(\log n)$ ...

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111 views

### Fractional Sobolev spaces on the circle with a Littlewood-Paley characterisation

Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L_p(\mathbb R)$.
Here, $F$ denotes the ...

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112 views

### A case of nested central limits

Consider the random variable $S=(s_0, \dots ,s_{N-1})$, a sequence of signs uniformly distributed on the hypercube $\{-1,1\}^N$. We are interested in $N$ large and prime. The Fourier transform ...

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163 views

### Statistical independence and the Fourier transform

Consider the random variable $S=(s_0, \dots ,s_{N-1})$, a sequence of signs uniformly distributed on the hypercube $\{-1,1\}^N$. With the Fourier transform we can define $N$ random walk variables
$$
...

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80 views

### Help with notations from 2D to 3D FFT representations as 1D FFT

I have this question on mathematics forum too, Notations, I thought of posting here, which ever place I get an answer, I will try to close it in the other.
I need some help and clarifications for my ...

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70 views

### Is there a space in which the $\vec a$ in $\sin(a_1\cdot x)+\sin(a_2\cdot x)$ is linear?

Suppose one has equations of the form $\sin(a_1\cdot x_i)+\sin(a_2\cdot x_i)=y_i$ for $i = 1, \dots, n$ (there are also amplitudes and phase shifts, but let's ignore these for now). I want to solve ...

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379 views

### Proof of a Fourier pair with Bessel functions?

How can we prove that the Fourier transform of the function
$$
f(x)
=
\begin{cases}
(a^2-x^2)^{c/2} BesselJ[c,b\sqrt{a^2-x^2}] & \text{for }x^2 < a^2\\
0 & \text{otherwise}
\end{cases}
$$
...

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120 views

### Simplifying an expression using tools from Fourier transform

Can anyone simplify the following expression? I guess something from Fourier transform can help:
$f(\omega) = \lim_\limits{R \to \infty} \frac{1}{R^2} \int_{r=0}^{R}{re^{ \omega r^{-\gamma}}} ...

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66 views

### Finite Ramanujan expansions over a finite field

I'm wondering if we could have an analogy in finite fields. After all, the Discrete Fourier Transform (DFT) has been generalized to finite fields as well (with essentially identical properties as in ...

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566 views

### Roots of characteristic function of “reciprocal gamma measure”

Let us call a measure $\mu$ on the Borel $\sigma$-algebra $\mathfrak{B}_{(0,\infty)}$ of subsets of $(0,\infty)$ a reciprocal gamma measure if it is absolutely continuous with respect to the Lebesgue ...

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148 views

### Generalization of Pitt's theorem

Pitt's theorem (Pitt 1937), states that the one-dimensional Fourier tranform is well defined and continuous between the weighted spaces $L^p(\mathbb{R},|x|^{bp}dx)$ and $L^q(\mathbb{R},|x|^{\beta ...

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70 views

### Relating the R-transform in free probability to noncommutative group representations

In traditional (commutative) probability theory, sums of random variables correspond to convolutions of distribution functions, which plays well with the Fourier Transform.
In free (noncommutative) ...

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132 views

### Isbell duality in Joyal and Street's Introduction to Tannaka Duality

In Sec. 3 of Joyal and Street's Introduction to Tannaka Duality and Quantum Groups, the authors give a commutative triangle of isomorphisms of compact topological groups (Corollary 8). This diagram ...

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183 views

### Fourier series and transform related to Epicycles

Let $\gamma:\mathbb{R}\to\mathbb{C}$ be a continuous periodic curve having a bounded variation.
1) Is it true that one can find a sequence of numbers $(r_n)_{n\in\mathbb{N}}\subset (0,\infty)$ and ...

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73 views

### Fourier transform of the realization of a stationary process in the space of tempered distributions?

A path of a stationary sequence of random variables $y_t$ does not have a discrete-time Fourier transform in the classical sense because it is not summable. This leads to considering the spectral ...

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159 views

### How are the infinity norm of Fourier transforms of sign vectors distributed?

This is a follow up to an earlier resolved question. Define the $n$-dimensional discrete Fourier transform via the matrix
$$
D_{s,t} := \omega^{st},
$$
where $\omega=\exp(-2\pi i/n)$. Notice that $D$ ...

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3k views

### Eliminating Gibbs phenomenon, and approximating with jumping functions in Fourier Analysis : An attempt and a question in this regard

This problem seems like a nightmare to me. I tried to expand
$K_{\omega}^f(t)$, but I am clueless of getting some kind of a closed
form or some kernel like structure. If I try to take ...

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92 views

### Real interpolation space between the Wiener algebra and $L^2$

The Wiener algebra $W_n$ is the image by the Fourier transform of $L^1(\mathbb R^n)$. What is the (complex) interpolation space between $W_n$ and $L^2(\mathbb R^n)$? It is probably not true that for ...

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99 views

### Inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$

Consider the inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$. My question is, how can we conclude about the decay properties, support and smoothness of the inverse Fourier transform? I ...

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116 views

### Technical question about a Fourier transform

I would like to know if there is an explicit expression for the Fourier transform of the following function:
$$f(x)=\mathbb{1}_{(0,\infty)}e^{-x-ix^2},$$
or to know where I can find some techniques to ...

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135 views

### Problem with operator and Fourier transform

I am currently dealing with a problem in functional analysis where I want to show the following.
Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$
if ...

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37 views

### Is Wiener amalgam spaces $W^{2,1}(\mathbb R)\subset C_0(\mathbb R)$?

I have been learning Wiener amalgam spaces.
In Wiener amalgam spaces $W(X, L^2)$, I am taking $X=\mathcal{F}L^{1}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}\},$ and $m(x)=1.$
Take $f(x)= ...

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187 views

### Finite trigonometric polynomial

I noticed by numerical and some explicit calculations for a few examples that for real-valued finitely supported functions $\phi \in L^2(\mathbb{R})$ we have that
$T(x):= \sum_{n \in \mathbb{Z}} ...

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242 views

### Proof of Biot-Savart theorem in 3D using distributions and fourier transform

While studying the book Vorticity and Incompressible Flow by A. Majda and A. Bertozzi I came across the following Biot-Savart theorem in 3D:
The solution to:
$$div \ v=0$$
$$curl(v)=\omega$$
is
...

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161 views

### Characterizations of Wiener algebra

The Wiener algebra $\mathcal W$ is defined as $\text{Fourier}(L^1(\mathbb R))$, i.e. the image by the Fourier transform of $L^1(\mathbb R)$. Riemann-Lebesgue's lemma ensures that
$$
\mathcal W\subset ...

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### Can we choose $g$ so that $\|(g\widehat{(f^{3})})^{\vee}\|_{L^{p}} \leq C \|g_{1}f\|_{L^{2}}^{r} \|(g_{2}\hat{f})^{\vee}\|_{L^{s}}$?

[I have asked this question on S.E. M; but I have not got any answer; and hope this is o.k. for M.O]
Let $f, f^{2}, f^{3}\in L^{q}(\mathbb R)\cap C_{0}(\mathbb R)$ where $ q\geq p, \ \text{and}$ and ...